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Properties

Label 2-175-5.4-c7-0-3
Degree $2$
Conductor $175$
Sign $0.894 + 0.447i$
Analytic cond. $54.6673$
Root an. cond. $7.39373$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 14.6i·2-s + 54.7i·3-s − 86.1·4-s − 801.·6-s − 343i·7-s + 612. i·8-s − 815.·9-s − 6.47e3·11-s − 4.71e3i·12-s + 1.16e4i·13-s + 5.01e3·14-s − 1.99e4·16-s + 1.34e4i·17-s − 1.19e4i·18-s − 3.49e4·19-s + ⋯
L(s)  = 1  + 1.29i·2-s + 1.17i·3-s − 0.672·4-s − 1.51·6-s − 0.377i·7-s + 0.423i·8-s − 0.373·9-s − 1.46·11-s − 0.788i·12-s + 1.47i·13-s + 0.488·14-s − 1.22·16-s + 0.664i·17-s − 0.482i·18-s − 1.16·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(175\)    =    \(5^{2} \cdot 7\)
Sign: $0.894 + 0.447i$
Analytic conductor: \(54.6673\)
Root analytic conductor: \(7.39373\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{175} (99, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 175,\ (\ :7/2),\ 0.894 + 0.447i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.4691772042\)
\(L(\frac12)\) \(\approx\) \(0.4691772042\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 + 343iT \)
good2 \( 1 - 14.6iT - 128T^{2} \)
3 \( 1 - 54.7iT - 2.18e3T^{2} \)
11 \( 1 + 6.47e3T + 1.94e7T^{2} \)
13 \( 1 - 1.16e4iT - 6.27e7T^{2} \)
17 \( 1 - 1.34e4iT - 4.10e8T^{2} \)
19 \( 1 + 3.49e4T + 8.93e8T^{2} \)
23 \( 1 + 7.78e4iT - 3.40e9T^{2} \)
29 \( 1 - 2.21e5T + 1.72e10T^{2} \)
31 \( 1 + 2.32e4T + 2.75e10T^{2} \)
37 \( 1 + 4.22e5iT - 9.49e10T^{2} \)
41 \( 1 - 1.91e5T + 1.94e11T^{2} \)
43 \( 1 + 3.10e5iT - 2.71e11T^{2} \)
47 \( 1 + 2.40e5iT - 5.06e11T^{2} \)
53 \( 1 - 1.06e6iT - 1.17e12T^{2} \)
59 \( 1 + 4.51e5T + 2.48e12T^{2} \)
61 \( 1 + 8.31e5T + 3.14e12T^{2} \)
67 \( 1 - 2.26e6iT - 6.06e12T^{2} \)
71 \( 1 + 2.22e6T + 9.09e12T^{2} \)
73 \( 1 + 4.99e6iT - 1.10e13T^{2} \)
79 \( 1 - 2.72e6T + 1.92e13T^{2} \)
83 \( 1 - 6.38e6iT - 2.71e13T^{2} \)
89 \( 1 - 7.32e6T + 4.42e13T^{2} \)
97 \( 1 + 2.38e6iT - 8.07e13T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.43680745077416922800832362932, −10.85549075894255542290481289273, −10.42208863852688946851947795079, −9.064830808137545178437373181929, −8.265180891275219169559217624113, −7.08475537155636107006746575027, −6.10030667447000794786714013841, −4.82485912312417685325514718931, −4.20630078266436256923475806695, −2.34535579846977756801015972934, 0.12300511325538373080072329152, 1.16235573075835217919670434175, 2.38138020682766066257039338620, 3.07397692342558551037510063975, 4.91231442150107180521269068031, 6.26625479956127827579251344849, 7.52110851168903530837009294803, 8.345260973671947250252093844354, 9.883667352151273925243704374136, 10.59910036649227003460681984197

Graph of the $Z$-function along the critical line